Reconciling scaling of the optical conductivity of cuprate superconductors with Planckian resistivity and specific heat

Materials tuned to a quantum critical point display universal scaling properties as a function of temperature T and frequency ω. A long-standing puzzle regarding cuprate superconductors has been the observed power-law dependence of optical conductivity with an exponent smaller than one, in contrast to T-linear dependence of the resistivity and ω-linear dependence of the optical scattering rate. Here, we present and analyze resistivity and optical conductivity of La2−xSrxCuO4 with x = 0.24. We demonstrate ℏω/kBT scaling of the optical data over a wide range of frequency and temperature, T-linear resistivity, and optical effective mass proportional to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim \ln T$$\end{document}~lnT corroborating previous specific heat experiments. We show that a T, ω-linear scaling Ansatz for the inelastic scattering rate leads to a unified theoretical description of the experimental data, including the power-law of the optical conductivity. This theoretical framework provides new opportunities for describing the unique properties of quantum critical matter.


REVIEWER COMMENTS
Reviewer #1 (Remarks to the Author): The manuscript titled "Planckian Behavior of Cuprate Superconductors: Reconciling the Scaling of Optical Conductivity with Resistivity and Specific Heat" by B. Michon et al. demonstrated that the power-law dependence of the optical conductivity with an exponent smaller than one is consistently connected with the <i>T</i>-linear dependence of the resistivity and the frequency (omega)-linear dependence of the optical scattering rate using the measured optical spectrum of LSCO near the pesudogap quantum critical point and a simple <i>T</i>, omega-linear Ansatz for the inelastic scattering rate. This manuscript was well-organized and well-written. It contains an interesting issue on the temperature-and frequency-dependent behaviors of charge carriers in strongly correlated electron systems, particularly copper oxide superconductors. However, there are a couple of important points that should be addressed before publication in Nature Communications.
1. In Fig. 1(d), the effective mass m*/m shows ~1.7 at near 0.4 eV, which is quite large compared with the asymptotic value, 1.0. The authors mentioned that the high value of m*/m at near 0.4 eV is caused by interband and/or mid-infrared transitions in LSCO. The authors need to show that it is true by either referring to appropriate literature (if there is) or using their own calculations. The interband and/or mid-infrared transitions would contribute to the optical scattering rate (h/τ) as well. 3. In the "Theory" section, the authors started from the Plankian Anzatz with ν = 1.0 and then calculated various quantities to compare with the experimental results. They also introduced a new parameter ν* and showed that the new parameter is dependent on the inelastic coupling constant (g). How is the parameter (ν*) related to the ν variable, which is 1.0 in the model? The relationship might be interesting.
4. The authors claimed that the Plankian model was better than the Sub-Plankian model for simulating the experimental results. But, I wonder whether the authors can completely rule out the Sub-Planckian model (ν < 1). The overall results (Extended Data Fig. 9) obtained using the Sub-Planckian are not so bad compared with the experimental results; only Extended Data Fig.  9(c) does not look so good for the experimental results. Is there a better parameter set for improving the Sub-Planckian results? As shown in Extended Data Fig. 2, ν and ε<sub>∞</sub> can be critical adjustable parameters.
5. I am just curious whether the Super-Planckian model (ν > 1) is possible. It might be helpful if the authors can give a comment on it.

Reviewer #3 (Remarks to the Author):
The authors have addressed all the remarks from my previous report. Therefore, I support the publication of this paper. One minor point is that Refs. 19 and 63 are now published. the power-law dependence of the optical conductivity with an exponent smaller than one is consistently connected with the T-linear dependence of the resistivity and the frequency (omega)-linear dependence of the optical scattering rate using the measured optical spectrum of LSCO near the pesudogap quantum critical point and a simple T, omega-linear Ansatz for the inelastic scattering rate. This manuscript was well-organized and well-written. It contains an interesting issue on the temperature-and frequencydependent behaviors of charge carriers in strongly correlated electron systems, particularly copper oxide superconductors. However, there are a couple of important points that should be addressed before publication in Nature Communications. Authors: We are pleased with the referee's positive overall assessment of our manuscript.
Reviewer #1: 1. In Fig. 1(d), the effective mass m*/m shows ~1.7 at near 0.4 eV, which is quite large compared with the asymptotic value, 1.0. The authors mentioned that the high value of m*/m at near 0.4 eV is caused by interband and/or mid-infrared transitions in LSCO. The authors need to show that it is true by either referring to appropriate literature (if there is) or using their own calculations. The interband and/or mid-infrared transitions would contribute to the optical scattering rate (h/τ) as well. Authors: The values of m*/m and h/ are both directly proportional to the spectral weight K used for converting the dielectric function data to conductivity data [see Eq. (14)]. The spectral weight of the Drude component is about 125 meV and the integration up to 2 eV of the total (Drude + mid-infrared) optical conductivity gives K = 211 meV, which is the value that we used. In the absence of other contributions to the optical conductivity, the  →  limit of the generalized Drude expression gives m*()/m → 1, whereas m*()/m ~ 211/125 in the intermediate energy range between the Drude and mid-infrared component, provided that these two components are well separated. Indeed, the data displayed in Fig. 1(d) show that the ratio m*()/m > 1. The fact that this quantity decreases strongly as a function of increasing energy is a consequence of the fact that the Drude and mid-infrared components of these experimental data are in fact not well separated.
Authors: Since this question was posed by a previous referee, we restate our answer. As our goal is to determine unknown parameters by optimizing the scaling collapse, it is logical to include in the fitting as much as possible of the data that scale and as few as possible of the data that do not scale. Figure 1c shows that 1/ appears to scale well up to 0.4 eV; there is a weak feature around 0.22 eV corresponding to the one seen in m*/m, but this does not break the scaling, as seen in Extended Data Fig.2. As a result, the  determined from 1/ is not very sensitive to the cutoff. On the contrary, it is very sensitive for m*/m, as explained in Supplementary Information Sec. B. Therefore, we have included the 1/ data between 0.22 and 0.4 eV, but not the m*/m data. In the revised Fig. 2, we show with dotted lines the excluded m*/m data. We also point out in Supplementary Information Sec. B that this choice is not essential: fitting all data below 0.22 eV or all data below 0.4 eV, we obtain very similar  = 2.91 and 3.03, respectively, and virtually the same values of m*(0). Reviewer #1: 3. In the "Theory" section, the authors started from the Planckian Ansatz with ν = 1.0 and then calculated various quantities to compare with the experimental results. They also introduced a new